How can I save two addresses for a customer with two different offices?

Yes, through multi aliases and sub ledger option you can enter two different addresses for a customer in BUSY.
Please follow the steps below for Multiple Aliases:
Step 1: Go to administration menu
Step 2: Click on configuration
Step 3: Features/options and then click on Account Tab
Step 4: Then click on maintain multiple account aliases option & also tick specify different address with multiple alias from multiple aliases configuration & save it and press esc button
Step 5: Then click on master option of administration menu 
Step 6: Click on Accounts option & then click on add / modify option
Step 7: After pressing enter on alias field then you will received multiple aliases window to enter separate address with multiple aliases of a party and then on same window you can enter different address of a party. 
Please follow the steps below for Sub Ledger:
Step 1: Go to administration menu
Step 2: Click on configuration
Step 3: Features/options and then click on Account Tab
Step 4: Click on maintain Sub Ledgers feature and save it
Step 5: Go back to Administration main menu and click on masters option
Step 6: Click on Accounts option & then click on add and then add party master with head office details and specifying its Ledger Type as General Ledger in party master and save.
Step 7: Create one more account with another/ reset address of a party & sspecifying its Ledger Type as Sub Ledger and selecting the General Ledger for it. (which party master we create with general ledger type). and then you will get combine data of both party master.